Choose the correct answer from the given four option.The differen… - Vidyadip

MCQ

Choose the correct answer from the given four option.The differential equation $\text{y}\frac{\text{d}\text{y}}{\text{d}\text{x}}+\text{x}=\text{C}$ represents:

A
Family of hype.
B
Family of parabolas.
C
Family of ellipses.
✓
Family of circles.

✓

Answer

Correct option: D.

Family of circles.

Given that, $\text{y}\frac{\text{d}\text{y}}{\text{d}\text{x}}+\text{x}=\text{C}$
$\Rightarrow\text{y}\frac{\text{d}\text{y}}{\text{d}\text{x}}=\text{C}-\text{x}$
$\Rightarrow\text{ydy}=(\text{C}-\text{x})\text{dx}$
On integrating both sides, we get
$\int\text{ydy}=\int(\text{C}-\text{x})\text{dx}$
$\Rightarrow\frac{\text{y}^2}{2}=\text{Cx}-\frac{\text{x}^2}{2}+\text{k}$
$\Rightarrow\frac{\text{x}^2}{2}+\frac{\text{y}^2}{2}=\text{Cx}+\text{k}$
$\Rightarrow\frac{\text{x}^2}{2}+\frac{\text{y}^2}{2}-\text{Cx}=\text{k}$
which represent family of circles.

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